where \(\log \)  is the logarithmic map on the manifold. See fig:discretecconcave for an illustration of a discrete \(c\) -concave function.
Intuitively, the optimal transport generated by discrete \(c\) -concave functions is piecewise constant as \(\exp _x(- \nabla _x \phi (x)) = \exp _x(-(-\log _x(y_{i_\star }))) = y_{i_\star }\) . This can be seen as generalizing semi-discrete optimal transport [1]}, which aims at finding transport maps between continuous and discrete probability measures, to the manifold setting.
